Catenaries, Parabolas and Suspension Bridges
David Griffin, "Catenaries, Parabolas and Suspension Bridges"
David Griffin, "Catenaries, Parabolas and Suspension Bridges"
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CATENARIES, PARABOLAS AND SUSPENSION BRIDGES<br />
DAVID GRIFFIN<br />
SUSPENSION BRIDGES / D A GRIFFIN
TALK CONTENTS<br />
A Brief, Pictorial History of <strong>Suspension</strong> <strong>Bridges</strong>.<br />
‣ Famous Examples<br />
‣ Problems <strong>and</strong> Solutions.<br />
Galileo<br />
‣ The Equations for the Chain Curve of a <strong>Suspension</strong> Bridge.<br />
‣ The Equation for a Hanging Chain. Parabola or Catenary?<br />
The Catenary <strong>and</strong> Jakob Bernoulli’s Challenge.<br />
‣ Leibniz<br />
‣ Hugens<br />
‣ Bernoulli<br />
The Methods for Deriving the Equation for the Catenary.<br />
‣Calculus.<br />
‣Differential Equation.<br />
‣The Calculus of Variations.<br />
The Relationship Between the Parabola <strong>and</strong> the Catenary.<br />
The Inverted Parabola.<br />
‣Arch <strong>Bridges</strong><br />
The Inverted Catenary.<br />
‣Arches<br />
‣Can <strong>Catenaries</strong> Help You to Cycle with Square Wheels?<br />
SUSPENSION BRIDGES / D A GRIFFIN
A “Suspended-deck <strong>Suspension</strong> Bridge”<br />
SUSPENSION BRIDGES / D A GRIFFIN
The remains of the Maya Bridge at Yaxchilan, Mexican/Guatemalan border.<br />
The earliest known suspension-deck suspension bridge. 100m in three spans. 7 th Century.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Faust Vrančić<br />
In 1595 the Croation bishop Faust Vrančić designed a<br />
suspension bridge, but it was never constructed.<br />
SUSPENSION BRIDGES / D A GRIFFIN
James Creek <strong>Suspension</strong> Bridge, Pennsylvania.<br />
James Finlay, 1801.<br />
Bridge demolished 1833.<br />
The first modern suspension bridge.<br />
It used wrought-iron cables.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Union Bridge, River Tweed, 1820.<br />
The oldest suspension bridge still<br />
carrying traffic.<br />
Dryburgh Bridge, River Tweed.<br />
Opened1817. Collapsed 1818.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Menai <strong>Suspension</strong> Bridge<br />
Thomas Telford, Completed 1826<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Clifton <strong>Suspension</strong> Bridge<br />
I K Brunel, Completed1864.<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
Two cables<br />
support one<br />
deck.<br />
SUSPENSION BRIDGES / D A GRIFFIN
BROOKLYN BRIDGE<br />
John Augustus Roebling,1883.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Four cables, two decks.<br />
SUSPENSION BRIDGES / D A GRIFFIN
A light walkway is suspended between the two decks of the Brooklyn Bridge.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Golden Gate <strong>Suspension</strong> Bridge<br />
Irving Morrow, Charle Alton Ellis, Leon Moissieff, 1937.<br />
SUSPENSION BRIDGES / D A GRIFFIN
PC<br />
LT<br />
Tacoma Narrows Bridge<br />
Leon Moissieff. Opened1940.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Map<br />
New Tacoma Narrows Bridge (Background: Mt Rainier)<br />
Charles E Andrew <strong>and</strong> Dexter R Smith,1950<br />
SUSPENSION BRIDGES / D A GRIFFIN
A second bridge at Tacoma Narrows was built in 2007.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Severn Bridge<br />
William Brown, 1966<br />
SUSPENSION BRIDGES / D A GRIFFIN
Brown designed the Severn Bridge to avoid<br />
the problems of the Tacoma Bridge. It has<br />
a slender, aerodynamic deck.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Humber Bridge<br />
John Hyatt, Douglas Strachan <strong>and</strong> others, 1981.<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE TOP TEN SUSPENSION BRIDGES<br />
<strong>Suspension</strong> bridge are typically ranked by the length of their main span.<br />
Akashi-Kaikyo Bridge (Japan) 1991 m 1998<br />
Xihoumen Bridge (China) 1650 m 2007<br />
Great Belt Bridge (Denmark) 1624 m 1998<br />
Runyang Bridge (China) 1490 m 2005<br />
Humber Bridge (Engl<strong>and</strong>) 1410 m 1981<br />
(The longest span from 1981 until 1998.)<br />
Jiangyin <strong>Suspension</strong> Bridge (China) 1385 m 1997<br />
Tsing Ma Bridge (Hong Kong), 1377 m 1997<br />
(Longest span with both road <strong>and</strong> metro.)<br />
Verrazano-Narrows Bridge (USA) 1298 m 1964<br />
(The longest span from 1964 until 1981.)<br />
Golden Gate Bridge (USA) 1280 m 1937<br />
(The longest span from 1937 until 1964.)<br />
Yangluo Bridge (China) 1280 m 2007<br />
SUSPENSION BRIDGES / D A GRIFFIN
Millenium Bridge 2000<br />
Arup, Foster <strong>and</strong> Partners<br />
PC<br />
LT<br />
Synchronous Lateral Excitation<br />
http://www2.eng.cam.ac.uk/~den/ICSV9_06.htm<br />
SUSPENSION BRIDGES / D A GRIFFIN
Dampers on Millenium Bridge to prevent synchronous lateral excitation.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The first person to study the physics <strong>and</strong> mathematics of the<br />
suspension bridge was Galileo.<br />
Galileo Galilei<br />
1564 - 1642<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGE: FORCES<br />
Cable<br />
y<br />
Tension at P.<br />
T<br />
P<br />
θ<br />
Tension at O.<br />
T o<br />
0<br />
Section of cable between O <strong>and</strong> P.<br />
P has horizontal coordinate x.<br />
Deck<br />
Weight of red<br />
section of deck.<br />
W<br />
Section of deck supported by<br />
cables segment OP. Length = x.<br />
SUSPENSION BRIDGES / D A GRIFFIN
y<br />
y<br />
T<br />
T<br />
T<br />
T o<br />
0<br />
P<br />
θ<br />
P<br />
θ<br />
x<br />
T o<br />
W<br />
x<br />
W<br />
The three forces T 0<br />
, T <strong>and</strong> W are in equilibrium.<br />
T<br />
W<br />
They form a triangle of forces with tanθ = W / T o<br />
.<br />
θ<br />
T is tangential to the chain at P.<br />
T o<br />
SUSPENSION BRIDGES / D A GRIFFIN
y<br />
y<br />
T<br />
T<br />
T<br />
W<br />
T o<br />
0<br />
P<br />
θ<br />
P<br />
θ<br />
x<br />
θ<br />
T o<br />
W<br />
x<br />
δy<br />
θ<br />
The triangle of forces is similar to the differential<br />
triangle at P.<br />
They both have gradient tanθ = W / T o<br />
.<br />
δx<br />
SUSPENSION BRIDGES / D A GRIFFIN
δy<br />
T<br />
W<br />
θ<br />
θ<br />
Gradient<br />
y<br />
<br />
x<br />
δx<br />
Gradient <br />
W<br />
T O<br />
T o<br />
y <br />
x<br />
W<br />
T O<br />
x = horizontal distance from the point O.<br />
W xg<br />
mass / length<br />
y<br />
xg<br />
<br />
x<br />
T O<br />
g<br />
k <br />
T O<br />
y kx<br />
x<br />
In the limit:<br />
dy <br />
dx<br />
kx<br />
μ has the dimensions of mass/length: m.l -1<br />
g has the dimensions of acceleration: l.t -2<br />
T o<br />
has the dimensions of force: m.l.t -2<br />
Thus k has the dimensions of length -1 : l -1<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE EQUATION FOR THE CHAIN-CURVE OF A SUSPENSION BRIDGE<br />
dy <br />
<br />
dx<br />
kx<br />
2<br />
kx<br />
y kx. dx<br />
y C 2<br />
Since y = 0 when x = 0 C must = 0<br />
2<br />
kx<br />
y <br />
2<br />
A Parabola<br />
k has the dimensions of length -1 .<br />
x has the dimensions of length.<br />
Thus:<br />
y has the dimensions of length.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Tension in the Cable<br />
T<br />
μxg<br />
θ<br />
T o<br />
T<br />
<br />
2 2 2<br />
g<br />
x <br />
T O<br />
The tension in the cable is greatest at the towers.<br />
The tension in the cable is a minimum at the lowest point <strong>and</strong> = T o<br />
.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Other Applications of <strong>Suspension</strong>-bridge Technology<br />
BRASSIERES:<br />
An Engineering Miracle<br />
From Science <strong>and</strong> Mechanics, February, 1964<br />
By Edward Nanas<br />
“There is more to brassiere design than meets the eye. In many respects, the challenge of<br />
enclosing <strong>and</strong> supporting a semi-solid mass of variable volume <strong>and</strong> shape, plus its adjacent<br />
mirror image - together they equal the female bosom - involves a design effort comparable to that<br />
of building a bridge or a cantilevered skyscraper. “<br />
http://www.firstpr.com.au/show-<strong>and</strong>-tell/corsetry-1/nanas/engineer.html<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE CATENARY<br />
Catenary, Alysoid, Chainette.<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
A spider’s web: multiple catenaries.<br />
SUSPENSION BRIDGES / D A GRIFFIN
“Simple suspension bridges”<br />
or rope bridges are catenaries,<br />
not parabolas.<br />
Söderskär Bridge, Finl<strong>and</strong>.<br />
SUSPENSION BRIDGES / D A GRIFFIN
GALILEO AND THE CATENARY<br />
Galileo believed that a catenary had the equation of a parabola.<br />
He had studied the parabola in various contexts <strong>and</strong> was the first<br />
to state that a projectile would follow the path of a parabola.<br />
In 1669 a posthumous publication by Joachim Jungius proved that<br />
the function describing a catenary could not be algebraic <strong>and</strong> could<br />
not therefore be a parabola.<br />
Joachim Jungius<br />
1587 – 1657.<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE JAKOB BERNOULLI CHALLENGE.<br />
In 1690 Jakob Bernoulli issued a challenge to Leibniz, Christiaan Huygens<br />
<strong>and</strong> Johann Bernoulli to derived the equation for the catenary.<br />
The solutions were presented in 1691.<br />
Gottfried Leibniz<br />
Christiaan Huygens<br />
Johann Bernoulli<br />
Newton also solved the<br />
problem: anonymously.<br />
The Age of Big Hair.<br />
Jakob Bernoulli<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Bernoulli Family Tree<br />
Several generations of<br />
mathematical geniuses.<br />
Jakob<br />
Johann (I)<br />
Daniel<br />
Nicolaus (II)<br />
Johann (II)<br />
Johann (III)<br />
Nicolaus (III)<br />
SUSPENSION BRIDGES / D A GRIFFIN
Jakob Bernoulli<br />
(1654-1705)<br />
First studied to be a minister.<br />
Studied at Basel University.<br />
‣ Received degree in theology.<br />
‣ Fascinated by mathematics.<br />
Furthered the calculus he had<br />
learned from Leibniz.<br />
Studied catenaries.<br />
Worked on the design of bridges.<br />
Studied the brachistochrone<br />
problem with Johann.<br />
Was a professor at Basel until<br />
his death.<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE SOLUTIONS TO THE BERNOULLI CHALLENGE.<br />
Leibniz used calculus, but did not show his method.<br />
Johann Bernoulli used the calculus of variations.<br />
‣This involves finding the shape which minimizes<br />
the potential energy of the system.<br />
Huygens used a complicated geometric proof.<br />
A solution using differential equations can also used.<br />
In 1691, when the derivation of the equation for the catenary was<br />
published, the Jesuit priest Ignace Gaston Paradies published a<br />
text-book on forces <strong>and</strong> geometry which included the derivation of<br />
the equations for the suspension-bridge cable <strong>and</strong> the catenary.<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
Cable<br />
y<br />
T<br />
Tension at P.<br />
P<br />
θ<br />
Tsinθ<br />
Tension at O.<br />
T o<br />
0<br />
s<br />
Tcosθ<br />
W = μsg<br />
SUSPENSION BRIDGES / D A GRIFFIN
Cable<br />
y<br />
T<br />
Tension at P.<br />
T<br />
P<br />
θ<br />
Tsinθ<br />
Tension at O.<br />
T o<br />
0<br />
s<br />
Tcosθ<br />
T o<br />
W = μsg<br />
µsg<br />
The three forces T 0<br />
, T <strong>and</strong> µsg are in equilibrium.<br />
They form a triangle of forces with tanθ = µsg / T o<br />
.<br />
Since T is tangential to the curve formed by the chain<br />
tanθ is equal to the gradient at the point P.<br />
θ<br />
T<br />
T o<br />
µsg<br />
Gradient = µsg / T o<br />
.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
We have shown:<br />
tan <br />
sg<br />
T O<br />
s<br />
<br />
TO<br />
g<br />
tan<br />
Define:<br />
k<br />
<br />
T<br />
O<br />
g<br />
T o<br />
has the dimension of force: m.l.t -2<br />
μ has the dimension of mass/length: m.l -1<br />
g has the dimension of acceleration: l.t -2<br />
Thus:<br />
k has the units of length: l<br />
s k tan<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
Consider the differential triangle.<br />
In the limit δs approaches the value<br />
of the hypotenuse.<br />
δS<br />
θ<br />
δy<br />
dx<br />
ds<br />
cos<br />
δx<br />
The above triangle is similar to the<br />
triangle of forces.<br />
T<br />
µsg<br />
θ<br />
T o<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
s k tan<br />
δS<br />
δy<br />
ds <br />
2<br />
k sec <br />
d<br />
(1)<br />
θ<br />
δx<br />
From (1) <strong>and</strong> (2):<br />
dx<br />
<br />
d<br />
dx<br />
ds<br />
<br />
ds<br />
d<br />
dx<br />
ds<br />
cos<br />
(2)<br />
dx<br />
<br />
d<br />
k.sec<br />
2<br />
.cos<br />
<br />
k sec<br />
dx <br />
d<br />
k sec<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
s k tan<br />
δS<br />
δy<br />
ds <br />
2<br />
k sec <br />
d<br />
(1)<br />
θ<br />
δx<br />
From (1) <strong>and</strong> (2):<br />
dy dy ds<br />
<br />
d ds d<br />
dy<br />
ds<br />
sin<br />
(2)<br />
dy <br />
d<br />
k sec 2 .sin<br />
dy <br />
d<br />
k sec.tan<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
Deriv<br />
dx <br />
d<br />
k sec<br />
<br />
<br />
x<br />
dx<br />
ksec .<br />
d<br />
Separate variables <strong>and</strong> integrate<br />
y<br />
x k.ln sec<br />
tan<br />
C<br />
T<br />
x<br />
<br />
0,<br />
<br />
0<br />
θ<br />
0 k.ln1<br />
0 C C 0<br />
P<br />
0<br />
x<br />
k. ln sec tan<br />
x<br />
Parametric equation of the catenary (1)<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
dy <br />
d<br />
k sec.tan<br />
<br />
<br />
y dy k sec.tan.<br />
d<br />
Separate variables <strong>and</strong> integrate<br />
y<br />
<br />
k sec<br />
C<br />
We have not defined where the axis y = 0 is.<br />
Define C = 0.<br />
Thus when θ = 0 y = k.<br />
}<br />
k<br />
x<br />
y <br />
k sec<br />
Parametric equation of the catenary (2)<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
We now have the two parametric equations for the catenary.<br />
We need to eliminate θ to obtain the cartesian x-y equation.<br />
x<br />
k. ln sec tan<br />
y k sec<br />
(1) (2)<br />
y<br />
From (2): sec (3)<br />
k<br />
2<br />
1<br />
tan sec<br />
2<br />
<br />
tan<br />
<br />
sec<br />
2<br />
1<br />
(4)<br />
From (1), (3) <strong>and</strong> (4):<br />
x<br />
<br />
k.ln<br />
y<br />
k<br />
<br />
<br />
<br />
<br />
y<br />
k<br />
<br />
<br />
<br />
2<br />
1<br />
1<br />
2<br />
Recall the identity: cosh u ln u u 1<br />
SUSPENSION BRIDGES / D A GRIFFIN
Derivation of the Catenary Equation Method 1<br />
2<br />
y y <br />
1<br />
2<br />
x k.ln<br />
1<br />
cosh u ln u u 1<br />
(1)<br />
k k <br />
(2)<br />
From (1) <strong>and</strong> (2):<br />
x<br />
<br />
k cosh<br />
1<br />
<br />
<br />
<br />
y<br />
k<br />
<br />
<br />
<br />
Rearranging <strong>and</strong> taking the<br />
cosh function of both sides<br />
of the equation gives:<br />
<br />
cosh<br />
<br />
x <br />
<br />
k <br />
<br />
y<br />
k<br />
y<br />
<br />
<br />
k cosh<br />
<br />
x <br />
<br />
k <br />
If x has the dimensions of length:<br />
‣ x/k is dimensionless.<br />
‣ y has the dimensions of length.<br />
Jungius was correct. The catenary is not described<br />
by an algebraic function; <strong>and</strong> is thus not a parabola.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Derivation of the Catenary by Differential Equations.<br />
We have previously shown:<br />
δS<br />
θ<br />
δx<br />
δy<br />
From (1) <strong>and</strong> (2):<br />
s k tan<br />
(1)<br />
dy<br />
tan (2)<br />
dx<br />
dy<br />
s k.<br />
dx<br />
2<br />
ds d y k. dx<br />
2<br />
dx<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Derivation of the Catenary by Differential Equations.<br />
δS<br />
θ<br />
δx<br />
δy<br />
ds 2 dx 2<br />
dy 2<br />
ds<br />
dx<br />
<br />
dy <br />
1<br />
<br />
dx <br />
2<br />
(1)<br />
We have just shown:<br />
ds <br />
dx<br />
k<br />
2<br />
d y<br />
. dx<br />
2<br />
(2)<br />
From (1) <strong>and</strong> (2):<br />
2<br />
d y dy <br />
k. 1<br />
2<br />
<br />
dx dx <br />
2<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
The Derivation of the Catenary by Differential Equations.<br />
2<br />
d y dy <br />
k. 1<br />
2<br />
<br />
dx dx <br />
2<br />
(1)<br />
Let:<br />
y'<br />
<br />
dy<br />
dx<br />
(2)<br />
From (1) <strong>and</strong> (2):<br />
<br />
<br />
dy'<br />
k. 1<br />
y<br />
dx<br />
' 2<br />
Separate variables<br />
2 dx<br />
d y'<br />
<br />
1<br />
y'<br />
.<br />
k<br />
dx<br />
k<br />
<br />
d<br />
1<br />
y'<br />
<br />
y'<br />
2<br />
<br />
dx<br />
k<br />
<br />
<br />
d<br />
1<br />
y'<br />
<br />
y'<br />
<br />
2<br />
SUSPENSION BRIDGES / D A GRIFFIN
dx<br />
k<br />
<br />
<br />
d<br />
1<br />
y'<br />
<br />
y'<br />
<br />
2<br />
(1)<br />
Recall the st<strong>and</strong>ard integral: <br />
du 1<br />
1<br />
u<br />
2<br />
sinh<br />
u<br />
(2)<br />
<br />
From (1) <strong>and</strong> (2): y'<br />
C<br />
x<br />
k<br />
sinh 1 When x 0 y' 0<br />
<br />
Thus: C 0<br />
Take sinh function of both sides:<br />
x<br />
k<br />
<br />
<br />
sinh 1 y'<br />
<br />
x <br />
sinh<br />
y'<br />
k <br />
<br />
dy<br />
dx<br />
x <br />
sinh<br />
<br />
k <br />
Separate the variables<br />
y<br />
<br />
<br />
k cosh<br />
<br />
x<br />
k<br />
<br />
<br />
<br />
C<br />
x<br />
dy . dx<br />
k<br />
<br />
sinh<br />
<br />
<br />
Once again we can define the<br />
coordinate axes so that C = 0.<br />
y<br />
<br />
<br />
k cosh<br />
<br />
x <br />
<br />
k <br />
SUSPENSION BRIDGES / D A GRIFFIN
The Relationship Between<br />
the Parabola <strong>and</strong> the Catenary<br />
SUSPENSION BRIDGES / D A GRIFFIN
Online function plotter: http://www.mathe-fa.de/en#anchor<br />
A Comparison of a Parabola <strong>and</strong> a Catenary<br />
SUSPENSION BRIDGES / D A GRIFFIN<br />
y<br />
x <br />
2.cosh <br />
2<br />
2<br />
<br />
y <br />
x<br />
2<br />
2
The Relationship Between the Parabola <strong>and</strong> the Catenary.<br />
The MacLaurin Series for a Catenary<br />
k cosh<br />
x<br />
k<br />
cosh<br />
x<br />
k<br />
<br />
0<br />
k<br />
2n<br />
2n<br />
x<br />
.(2n)!<br />
2 4 6<br />
cosh x x x x<br />
1<br />
........<br />
2 4 6<br />
k k .2! k .4! k .6! k<br />
2<br />
2 4 6<br />
cosh x x x x<br />
k k(1<br />
........<br />
2 4 6<br />
k k .2! k .4! k .6! k<br />
2<br />
2n<br />
x<br />
........<br />
.(2n)!<br />
n<br />
2n<br />
x<br />
........)<br />
.(2n)!<br />
2 4 6<br />
2n<br />
cosh x x x x x<br />
k k ........ ........)<br />
3 5<br />
2n<br />
k k.2!<br />
k .4! k .6! k<br />
1 .(2n)!<br />
n<br />
k cosh<br />
x<br />
k<br />
2<br />
x<br />
2 k<br />
k<br />
If k is >1 the catenary can be approximated<br />
by a parabolic function for small values of x.<br />
SUSPENSION BRIDGES / D A GRIFFIN
If k is >1 the catenary can be approximated by a parabolic function for small values of x.<br />
(k = 2)<br />
x <br />
y 2.cosh <br />
2 <br />
y <br />
2<br />
x<br />
4<br />
2<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Relationship Between the Parabola <strong>and</strong> the Catenary<br />
If the parabola y = x 2 is rolled along the x-axis the locus<br />
of its focus is the catenary:<br />
1<br />
y cosh 4x<br />
4<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Construction of the Brooklyn Bridge<br />
Parabola <strong>and</strong> Catenary<br />
During the construction of the Brooklyn Bridge<br />
it was possible at one stage to contrast a laden<br />
<strong>and</strong> an un-laden cable: a parabola <strong>and</strong> a catenary.<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
Inverted <strong>Parabolas</strong> <strong>and</strong> <strong>Catenaries</strong><br />
Arch <strong>Bridges</strong><br />
Free-st<strong>and</strong>ing Arches<br />
SUSPENSION BRIDGES / D A GRIFFIN
The New River Gorge Bridge, Virginia.<br />
A supported-deck bridge.<br />
SUSPENSION BRIDGES / D A GRIFFIN
The Tyne Bridge<br />
A compression-arch suspended-deck bridge<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUPPORTED ARCH BRIDGE<br />
The forces acting on a section of the arch are compressive.<br />
An analysis of the triangle of forces leads once again to a parabola.<br />
The arch is rigid. I does not assume the shape of an inverted parabola.<br />
It should be constructed as an inverted parabola if it is to have a uniform<br />
deck supported at regular intervals.<br />
The triangle of forces acting on a segment is analogous to that for a suspension bridge.<br />
y<br />
W<br />
C O<br />
x<br />
C O<br />
θ<br />
C<br />
W = μxg<br />
C<br />
SUSPENSION BRIDGES / D A GRIFFIN
THE INVERTED CATENARY<br />
The inverted catenary is the ideal curve for<br />
an arch which supports only its own weight.<br />
It is the minimum energy structure.<br />
The forces are primarily of compression.<br />
SUSPENSION BRIDGES / D A GRIFFIN
.<br />
St Louis Gateway Arch,<br />
Eero Saarinen, Completed 1965.<br />
SUSPENSION BRIDGES / D A GRIFFIN
St Louis Gateway Arch<br />
SUSPENSION BRIDGES / D A GRIFFIN
This formula is inscribed on the arch.<br />
Thus when x = 0 <strong>and</strong> y is at a maximum<br />
y = 630 ft.<br />
This also gives a separation of 630 ft<br />
for the bases.<br />
SUSPENSION BRIDGES / D A GRIFFIN
SUSPENSION BRIDGES / D A GRIFFIN
Taq–i-Kisra, Ctesiphon, Mesopotamia / Irak<br />
SUSPENSION BRIDGES / D A GRIFFIN
Casa Milà, Barcelona.<br />
Antoni Gaudí<br />
SUSPENSION BRIDGES / D A GRIFFIN
Casa Milà, Barcelona.<br />
Antoni Gaudí<br />
SUSPENSION BRIDGES / D A GRIFFIN
Casa Milà, Barcelona.<br />
Antoni Gaudí<br />
SUSPENSION BRIDGES / D A GRIFFIN
Can you ride<br />
a cycle with<br />
square wheels?<br />
http://www.maa.org/mathl<strong>and</strong>/mathtrek_04_05_04.html<br />
SUSPENSION BRIDGES / D A GRIFFIN
For the rolling square the shape of the road is a series<br />
of inverted, truncated catenary curves.<br />
PC<br />
LT<br />
SUSPENSION BRIDGES / D A GRIFFIN
For regular n-sided polygonal wheels the curve of the road<br />
is made from inverted catenaries with the equation:<br />
y = - Rcot(/n).cosh(x/A)<br />
SUSPENSION BRIDGES / D A GRIFFIN
In practice a triangular wheel would get stuck.<br />
The vertices puncture the road.<br />
Road<br />
“Spoke”<br />
Bottom of triangular wheel<br />
SUSPENSION BRIDGES / D A GRIFFIN
www.exploratorium.edu/texnet/exhibits/motion/square.../square_cbk.pdf<br />
SUSPENSION BRIDGES / D A GRIFFIN
Proving that the locus of the centre of a square as it<br />
rolls over an inverted catenaries is a straight line.<br />
There are several, diverse proofs. Some are long <strong>and</strong> complicated.<br />
www.maa.org/pubs/mathmag.html<br />
(General study on “roads” <strong>and</strong> non-circular “wheels”.)<br />
www.macalester.edu/mathcs/documents/catenaries.pdf<br />
(Uses two coordinate systems: polar <strong>and</strong> cartesian.)<br />
http://www.maplesoft.com/applications/view.aspx?SID=6322<br />
(Method using several differential equations.)<br />
http://www.snc.edu/math/squarewheelbike.html<br />
Follow hypertext link on the website mathematics.<br />
(Uses st<strong>and</strong>ard geometry <strong>and</strong> calculus: but long.)<br />
See: Wikipedia page on Roulette (curve).<br />
(Generates the catenary road as a roulette in the complex plane.)<br />
SUSPENSION BRIDGES / D A GRIFFIN
Consider the square resting<br />
on a vertex <strong>and</strong> symmetrically<br />
poised between two of the<br />
humps.<br />
We want the centre of the<br />
circle to remain on the line<br />
y = a√2 as the square rolls<br />
along the road.<br />
y<br />
2a<br />
Y = a√2<br />
a√2<br />
Road, y = f(x)<br />
x<br />
SUSPENSION BRIDGES / D A GRIFFIN
y<br />
2a<br />
C<br />
θ<br />
y = a√2<br />
B<br />
θ<br />
Road, y = f(x)<br />
A<br />
x<br />
SUSPENSION BRIDGES / D A GRIFFIN
a.sec y a<br />
2<br />
y<br />
dy<br />
dx<br />
<br />
tan<br />
2a<br />
C<br />
θ<br />
y = a√2<br />
a.secθ<br />
a<br />
B<br />
θ<br />
Road, y = f(x)<br />
Appendix<br />
y<br />
A<br />
SUSPENSION BRIDGES / D A GRIFFIN<br />
x
a.sec y a<br />
2<br />
dy<br />
dx<br />
tan<br />
2 2<br />
1tan sec<br />
<br />
sec 2 <br />
2<br />
y<br />
a<br />
(1)<br />
dy<br />
2<br />
1<br />
sec<br />
<br />
dx<br />
<br />
(2)<br />
Let:<br />
u sec<br />
(3)<br />
From (1) <strong>and</strong> (3):<br />
u <br />
2 <br />
y<br />
a<br />
From (2) <strong>and</strong> (3):<br />
dy<br />
<br />
1<br />
<br />
dx<br />
<br />
2<br />
u<br />
2<br />
(4)<br />
dy<br />
dx <br />
u<br />
2<br />
1<br />
SUSPENSION BRIDGES / D A GRIFFIN
dy 2<br />
u 1<br />
dx u 2 <br />
y<br />
a<br />
du du dy<br />
<br />
dx dy dx<br />
du<br />
dy<br />
1<br />
<br />
a<br />
du<br />
dx<br />
1<br />
a<br />
2<br />
<br />
u<br />
1<br />
Separate variables.<br />
1<br />
dx <br />
a<br />
u<br />
du<br />
2<br />
1<br />
SUSPENSION BRIDGES / D A GRIFFIN
1<br />
dx <br />
a<br />
u<br />
du<br />
2<br />
1<br />
<br />
1<br />
dx <br />
a<br />
<br />
u<br />
du<br />
2<br />
1<br />
x<br />
a<br />
x <br />
u C<br />
<br />
a <br />
1<br />
C<br />
cosh<br />
u<br />
cosh<br />
u <br />
2 <br />
y<br />
a<br />
Take cosh function of both sides.<br />
x<br />
y a 2 a.cosh<br />
<br />
C<br />
<br />
a <br />
y x<br />
2 coshC<br />
<br />
a a<br />
When x = 0 y = f(x) = 0. Therefore:<br />
0 <br />
0a 2 a.coshC<br />
<br />
a <br />
<br />
2 cosh C<br />
<br />
C cosh 2<br />
1<br />
<br />
<br />
ya a<br />
<br />
a <br />
1<br />
2 cosh cosh 2 x<br />
<br />
The equation of the road.<br />
SUSPENSION BRIDGES / D A GRIFFIN
Plots of the Inverted Catenary “Road” <strong>and</strong> Related Functions.<br />
1<br />
x <br />
y a 2 acoshcosh 2 <br />
<br />
a <br />
Road equation<br />
y cosh( x)<br />
y <br />
y<br />
<br />
1<br />
cosh cosh 2<br />
<br />
x<br />
1<br />
2 cosh cosh 2 <br />
(Road equationtion with a = 1)<br />
<br />
x<br />
<br />
SUSPENSION BRIDGES / D A GRIFFIN
1<br />
y a 2 acoshcosh 2<br />
<br />
x <br />
<br />
a <br />
Solve for y = 0 <strong>and</strong> dy/dx = 0<br />
y<br />
(Values when a = 1)<br />
<br />
1<br />
a.cosh 2, a 2 a<br />
<br />
(0.8814, 0.4142)<br />
<br />
1<br />
2.cosh a 2,0<br />
<br />
(1.7627, 0)<br />
(0,0)<br />
x<br />
SUSPENSION BRIDGES / D A GRIFFIN
PC<br />
PC<br />
LT<br />
LT<br />
SUSPENSION BRIDGES / D A GRIFFIN
APPENDIX 1<br />
Proof that the Arc Length of the Inverted <strong>Catenaries</strong> is<br />
Equal to the Length of a Side of the Square: i.e. 2a.<br />
dx<br />
<br />
s <br />
1 . dy<br />
dy<br />
<br />
2<br />
dy<br />
tan<br />
dx<br />
dx<br />
cot<br />
dy<br />
(1) (2) (3)<br />
<br />
2<br />
From (1) <strong>and</strong> (3): s 1cot . dy<br />
cos .<br />
s ec dy<br />
(4)<br />
a.sec y a<br />
2<br />
y a 2 asec<br />
dy<br />
a sec .tan<br />
d <br />
(5)<br />
From (4) <strong>and</strong> (5):<br />
s a.cos ec .sec .tan .<br />
d<br />
2<br />
s a.sec .<br />
d<br />
<br />
<br />
SUSPENSION BRIDGES / D A GRIFFIN
y<br />
2a<br />
a√2<br />
45 o 45 o 135 o<br />
x<br />
SUSPENSION BRIDGES / D A GRIFFIN
2<br />
s a.sec .<br />
d<br />
<br />
<br />
2<br />
2<br />
s a.sec .<br />
d<br />
<br />
<br />
1<br />
<br />
2<br />
s a tan<br />
C <br />
1<br />
1 45 o<br />
2 135 o<br />
tan 45 1<br />
tan135 1<br />
s a1 1 2a<br />
<br />
<br />
SUSPENSION BRIDGES / D A GRIFFIN
APPENDIX 2<br />
TNB<br />
SUSPENSION BRIDGES / D A GRIFFIN
APPENDIX 3<br />
Conversion of FLV (e.g. Youtube) files to downloadable files (e.g. MP4).<br />
SUSPENSION BRIDGES / D A GRIFFIN
APPENDIX 4<br />
http://www.mathe-fa.de/en<br />
SUSPENSION BRIDGES / D A GRIFFIN
APPENDIX 5<br />
1 2<br />
cosh x ln x x 1<br />
e<br />
cosh x <br />
e<br />
sinh x <br />
x<br />
x<br />
e<br />
2<br />
e<br />
2<br />
x<br />
x<br />
cosh x sinh x e<br />
x<br />
2 2<br />
cosh x sinh x 1<br />
y<br />
cosh<br />
1<br />
x<br />
x<br />
cosh<br />
y<br />
cosh<br />
y<br />
sinh y e<br />
y<br />
2 2<br />
cosh y<br />
sinh y 1<br />
y<br />
2<br />
e cosh y cosh y1<br />
e x x<br />
y<br />
2<br />
<br />
1<br />
y x x <br />
2<br />
ln 1<br />
1 2<br />
cosh x ln x x 1<br />
SUSPENSION BRIDGES / D A GRIFFIN